The Shifted Convolution of Divisor Functions

TL;DR

This paper establishes an asymptotic formula for the shifted convolution of divisor functions, providing uniform results with power-saving error terms, and extends these results to Fourier coefficients of cusp forms, improving prior bounds.

Contribution

It introduces a new asymptotic formula for shifted divisor function convolutions with uniformity and power-saving error, also applying to cusp form coefficients, advancing previous work.

Findings

Derived an asymptotic formula with power-saving error for divisor function convolutions.

Extended the results to Fourier coefficients of holomorphic cusp forms.

Improved bounds over previous estimates in the literature.

Abstract

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of $d_3(n)$ and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.